Centered matrices

CenteredMatrices{𝔽,T} <: AbstractDecoratorManifold{𝔽}

The manifold of $m×n$ real-valued or complex-valued matrices whose columns sum to zero, i.e.

\[\bigl\{ p ∈ 𝔽^{m×n}\ \big|\ [1 … 1] * p = [0 … 0] \bigr\},\]

where $𝔽 ∈ \{ℝ,ℂ\}$.

Constructor

CenteredMatrices(m, n[, field=ℝ]; parameter::Symbol=:type)

Generate the manifold of m-by-n (field-valued) matrices whose columns sum to zero.

parameter: whether a type parameter should be used to store m and n. By default size is stored in type. Value can either be :field or :type.

Y = Weingarten(M::CenteredMatrices, p, X, V)
Weingarten!(M::CenteredMatrices, Y, p, X, V)

Compute the Weingarten map $\mathcal W_p$ at p on the CenteredMatrices M with respect to the tangent vector $X \in T_p\mathcal M$ and the normal vector $V \in N_p\mathcal M$.

Since this a flat space by itself, the result is always the zero tangent vector.

check_point(M::CenteredMatrices, p; kwargs...)

Check whether the matrix is a valid point on the CenteredMatrices M, i.e. is an m-by-n matrix whose columns sum to zero.

The norm of the column sums of p has to be at most atol, or at most rtol * sqrt(m) * norm(p) if this scale is finite. The relative tolerance rtol refers to the size of p; its default is the one of isapprox.

check_vector(M::CenteredMatrices, p, X; kwargs... )

Check whether X is a tangent vector to manifold point p on the CenteredMatrices M, i.e. that X is a matrix of size (m, n) whose columns sum to zero and its values are from the correct AbstractNumbers. The column sums of X have to vanish up to max(atol, rtol * sqrt(m) * norm(X)). The relative tolerance rtol refers to the size of X; its default is the one of isapprox.

get_coordinates(M::CenteredMatrices, p, X, ::DefaultOrthonormalBasis{ℝ})

Compute the coordinates of $HX$ in the Euclidean space of $(m-1)×n$ matrices, where $H$ is the $(m-1)×m$ Helmert submatrix, see [Gen17, Section 8.8.1].

get_vector(M::CenteredMatrices, p, c, ::DefaultOrthonormalBasis{ℝ})

Compute $X = H^{\mathrm{T}}Z$, where $Z$ is the $(m-1)×n$ matrix with the coordinates c in the Euclidean space and $H$ is the $(m-1)×m$ Helmert submatrix, see [Gen17, Section 8.8.1].

is_flat(::CenteredMatrices)

Return true. CenteredMatrices is a flat manifold.

manifold_dimension(M::CenteredMatrices)

Return the manifold dimension of the CenteredMatrices m-by-n matrix M over the number system 𝔽, i.e.

\[\dim(\mathcal M) = (m*n - n) \dim_ℝ 𝔽,\]

where $\dim_ℝ 𝔽$ is the real_dimension of 𝔽.

project(M::CenteredMatrices, p, X)

Project the matrix X onto the tangent space at p on the CenteredMatrices M, i.e.

\[\operatorname{proj}_p(X) = X - \begin{bmatrix} 1\\ ⋮\\ 1 \end{bmatrix} * [c_1 \dots c_n],\]

where $c_i = \frac{1}{m}\sum_{j=1}^m x_{j,i}$ for $i = 1, \dots, n$.

project(M::CenteredMatrices, p)

Projects p from the embedding onto the CenteredMatrices M, i.e.

\[\operatorname{proj}_{\mathcal M}(p) = p - \begin{bmatrix} 1\\ ⋮\\ 1 \end{bmatrix} * [c_1 \dots c_n],\]

where $c_i = \frac{1}{m}\sum_{j=1}^m p_{j,i}$ for $i = 1, \dots, n$.

riemann_tensor(M::CenteredMatrices, p, X, Y, Z)

Compute the Riemann tensor $R(X,Y)Z$ at point p on CenteredMatrices M. Its value is always the zero tangent vector.

sectional_curvature(M::CenteredMatrices, p, X, Y)

Sectional curvature of CenteredMatrices M is 0.

sectional_curvature_max(M::CenteredMatrices)

Return the upper bound 0 of the sectional curvature of CenteredMatrices M, which is flat.

sectional_curvature_min(M::CenteredMatrices)

Return the lower bound 0 of the sectional curvature of CenteredMatrices M, which is flat.

rand(M::CenteredMatrices; vector_at=nothing, σ::Real=1.0)
rand!(rng::AbstractRNG, M::CenteredMatrices, pX; vector_at=nothing, σ::Real=1.0)

Project a matrix of independent normally distributed entries with standard deviation σ onto M, which yields a random point as well as a random tangent vector at vector_at.